jpg vs transparent png

I create beautiful logos - just what i want - and save them. Then I export them as jpg's and the results are perfect. Then I save them as transparent png's to use on web sites - that's what the site owner wants. Trouble is, the transparent png's are awful. The white backgrounds of the underlying image files show through. (I have tried "removing white background" from the software without it being any help in the end result.) The transparent gifs are basically unusable. why is it the jpg's look perfect while the transparent png's don't?
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